Goodness of fit test

Used to test how well a proposed model (distribution) fits the observed data. The $χ^{2}$ Chi-Squared Test is often a common choice for carrying out a goodness-of-fit test

Step 1. Estimate the parameter for the assumed distribution

Step 2. Compute the probability for each observation under the assumed distribution

Step 3. Compute the expected frequencies

Step 4. Test the goodness-of-fit of the assumed distribution to the observed data

Examples

Number of errors	Observed frequencies $f_{i}$	Poisson probability with $λ = 3$	Expected frequencies $E_{i}$
0	18
1	53
2	103
3	107
4	82
5	46
6	18
7	10
8	2
9	1

Step 1. Estimate the parameter for the assumed distribution
$n = 440, \sum_{i} x_{i} = 1341$ , the number of errors observed in the sample

\hat{λ} = \bar{x} = \frac{\sum x_{i}}{n} = \frac{1341}{440} = 3.05

We test

H_{0} : X \sim P o i s s o n (3) v s H_{a} : X ≁ P o i s s o n (3)

Could indicate that the estimate given is not good, or that the distribution doesn't fit

Step 2. Compute the probability for each observation under the assumed distribution

P (x = x) = \frac{e^{- λ} λ^{x}}{x!} = \frac{e^{- 3} 3^{x}}{x!} under H_{0}, x = 0, \dots, 9

Number of errors	Observed frequencies $f_{i}$	Poisson probability with $λ = 3$
0	18	0.0498
1	53	0.1494
2	103	0.2240
3	107	0.1680
4	82	0.1008
5	46	0.0504
6	18	0.0216
7	10	0.0081
8	2	0.0081
9	1	0.0038

Step 3. Compute the expected frequencies

Let $E_{i j}$ be the expected # of observations that make $j$ errors among 440 observations

Number of errors	Observed frequencies $f_{i}$	Poisson probability with $λ = 3$	Expected frequencies $E_{i}$
0	18	0.0498	21.9
1	53	0.1494	65.7
2	103	0.2240	98.6
3	107	0.1680	98.6
4	82	0.1008	73.9
5	46	0.0504	44.4
6	18	0.0216	22.2
7	10	0.0081	9.5
8	2	0.0081	3.6
9	1	0.0038	1.7
10	0
...

based on the fischer exact test (?), the Chi-Squared Test will become less accurate for values $< 5$ , we can group all those groups together

Step 4.de
Test the goodness-of-fit of the Poisson distribution to the observed data

H_{0} : X \sim P o i s s o n (3) v s H_{a} : X ≁ P o i s s o n (3)

under $H_{0} :$

\begin{aligned} χ^{2} & = \sum_{i = 0}^{8} \frac{(f_{i} - E_{i})^{2}}{E_{i}} \sim χ_{m - t - 1}^{2} = χ_{7}^{2} \end{aligned}

$d f =$ number of groups - number of unknown parameters

Reject $H_{0}$ if

\begin{aligned} χ_{o b s}^{2} & \geq χ_{7, 0.95}^{2} = 14.067 \\ χ_{o b s}^{2} = 6.83 & \geq 14.067 \end{aligned}